Demonstrate that the process in which the work performed by an ideal gas is alphaortional to that corresponding increment of its internal energy is decribed by the equation `p V^n - const`, where `n` is a constant.

Explanation

According to the problem : `A alpha U` or `dA = aU` (where `a` is alphaortionality constant)
or, `pdV = (a v R dT)/(gamma - 1)`…(1) ltrgt From ideal gas law, `pV = v R T`, on differentiating
`pdV + Vdp = v RdT`…(2)
Thus from (1) and (2)
`pdV = (a)/(gamma - 1) (pdV + Vdp)`
or, `pdV ((a)/(gamma - 1) - 1) + (a)/(gamma - 1) V dp = 0`
or, `pdV(k - 1) + kVdp = 0`(where `k = (a)/(gamma - 1) = "another constant")`
or, `pdV (k - 1)/(k) + Vdp = 0`
or, `pdVn + Vdp = 0`(where `(k - 1)/(k) = n = ratio`)
Diving both the sides by `pV`
`n (dV)/(V)+(dp)/(p) = 0`
On integrating `n 1n V + 1 n p = 1n C` (where `C` is constant)
or, `1 n(pV^n) = 1n C` or, `pV^n = C (const)`.